An adaptive and efficient affine transformation-based subdivision method for evaluation of nearly singular integrals

Abstract

Based on the Affine transformation and Partitioning techniques, we present here an adaptive element Subdivision Method (APSM) for efficient evaluation of nearly singular integrals. Adaptive subdivision techniques can deal with the common situation where the size and shape of boundary elements are significantly different. We first introduce the basic structure and main ideas of APSM implementation via affine transformations, then present several different kinds of element subdivision results with arbitrary shapes. There are several advantages of the APSM over other element subdivision methods, including adaptive subdivision, improved accuracy, and simplicity of implementation. By means of affine transformations and partitioning techniques, it is possible to subdivide a given element into a set of projective and refinement zones. It is more flexible and convenient to perform the successful subdivision of the projective and refinement zones, respectively. In addition, the ultimate patch generation quality can be improved by incorporating certain types of boundary serendipity patches around the source point. With the introduction of these serendipity patches, the APSM is capable of considerably greater accuracy and efficiency for systematic computation of the integration scheme. Several numerical examples have been given to verify the effectiveness, feasibility and robustness of the illustrated integration schemes.